Download Problem Set 5 - Stony Brook Mathematics

MAT 539 Algebraic Topology
Problem Set 5
due Monday, December 13 (you can hand it in anytime on or before Dec 13)
Hatcher 3, 7 p.229 (the ring structure of H ∗ (RPn ) and H ∗ (CPn ) will be determined in
Monday class).
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Problem 1. Show that if X is a finite simplicial complex whose underlying topological
space is a homology n-manifold, then
(a) X consists entirely of n-simplices and their faces,
(b) Every (n − 1)-simplex is a face of precisely two n-simplices.
Problem 2. Suppose that X is a compact triangulable homology n-manifold.
(a) Show that if X is orientable, then Hn−1 (X) is torsion-free, and for any coefficient
group G, Hn (X; G) = G = H n (X; G).
(b) Show that if X is non-orientable, then the torsion subgroup of Hn−1 (X) is Z/2,
Hn (X; G) = ker(G →2 G), and H n (X; G) = G/2G.
In particular, Hn (X) = 0, H n (X) = Z/2.
Problem 3. Let X be a homology n-manifold (not necessarily compact) that is triagulated
by a locally finite simplicial complex. (“Locally finite” means that each vertex belongs
to finitely many simplices.) The Poincaré duality holds in this setting if one considers
cohomology with compact support Hcp (X; G). To define this for a simplicial complex K,
let Ccp (K; G) ⊂ Hom(Cp (K), G) be the group of homomorphisms that vanish on all but
finitely many oriented simplices of K.
(a) Show that if K is locally finite, then δ maps Ccp into Ccp+1 , thus the resulting cohomology groups Hcp (K; G) are well defined.
(b) If K is the complex whose total space is R and whose vertices are the integers, show
that Hc1 (K) = Z and Hc0 (K) = 0.
(c) Show that Hcp (X; Z/2) = Hn−p (X; Z/2).
(d) We say that X is orientable if it’s possible
to orient the n-simplices
σi of X such that
P
P
the resulting (possibly infinite) chain
σi is a cycle (i.e. ∂( σi ) = 0 when computed
formally). Show that in this case Hcp (X; G) = Hn−p (X; G).
Problem 4. Let X, Y be compact connected triangulable homology n-manifolds.
(a) Show that if X and Y are orientable, and there is a continuous map f : X → Y has
non-zero degree, then bi (X) ≥ bi (Y ), where bi stands for the i-th Betti number.
(b) Denote by Xg the orientable closed surface of genus g. Show that there exists a
continuous map f : Xn → Xm of non-zero degree if and only if n ≥ m.
Problem 5. Show that the Euler characteristic of a compact orientable triangulable
homology manifold of odd dimension is always zero.